Problem: Let $f$ be a scalar field. Is the expression $\nabla \cdot (\nabla f)$ a scalar field, a vector field, or undefined? Choose 1 answer: Choose 1 answer: (Choice A) A Scalar field (Choice B) B Vector field (Choice C) C Undefined
Answer: The gradient, which takes a scalar field and gives a vector field, can be written in two ways: $\text{grad}(f) = \nabla f$ The divergence, which takes a vector field and gives a scalar field, can also be written in two ways: $\text{div}(F) = \nabla \cdot F$ Therefore, $\nabla \cdot (\nabla f)$ is the divergence of the gradient of a scalar field. The gradient of a scalar field is a vector field. The divergence of a vector field is a scalar field. The expression $\nabla \cdot (\nabla f)$ is a scalar field.